Explanation:

In a linear elastic structural system, the potential energy is minimized to determine the equilibrium state of the system. This is achieved by minimizing the total potential energy of the system with respect to the unknown displacements. The minimization of potential energy leads to the derivation of certain conditions and relations that are essential for the analysis of the system.

a) Compatibility Conditions:
The minimization of potential energy requires the system to satisfy the compatibility conditions. These conditions ensure that the displacements and strains within the system are consistent and compatible with each other. The compatibility conditions relate the displacements at different points in the system and are necessary for the system to be in a state of equilibrium.

b) Constitutive Relations:
The minimization of potential energy also leads to the establishment of constitutive relations. These relations describe the behavior of the material used in the structural system. In a linear elastic system, the constitutive relations relate the stress and strain in the material. The constitutive relations are necessary to determine the internal forces and stresses within the system.

c) Equilibrium Equations:
The minimization of potential energy yields the equilibrium equations of the system. These equations ensure that the system is in a state of equilibrium, where the sum of the external forces acting on the system is balanced by the internal forces. The equilibrium equations can be derived by taking the partial derivatives of the potential energy with respect to the unknown displacements and setting them equal to zero.

d) Strain-Displacement Relations:
The minimization of potential energy also results in the derivation of strain-displacement relations. These relations relate the strains in the system to the displacements. In a linear elastic system, the strain-displacement relations are linear and can be expressed in terms of the displacement gradients. The strain-displacement relations are necessary to determine the strains within the system.

In conclusion, the minimization of potential energy in a linear elastic structural system yields compatibility conditions, constitutive relations, equilibrium equations, and strain-displacement relations. These conditions and relations are essential for the analysis and design of the structural system.